Trigonometric ratios
Trigonometry involves calculating angles and sides in triangles.
Labelling the sides
The three sides of a right-angled triangle have special names.
The hypotenuse (\(h\)) is the longest side. It is opposite the right angle.
The opposite side (\(o\)) is opposite the angle in question (\(x\)).
The adjacent side (\(a\)) is next to the angle in question (\(x\)).
Three trigonometric ratios
Trigonometry involves three ratios - sine, cosine and tangent which are abbreviated to \(\sin\), \(\cos\) and \(\tan\).
The three ratios are calculated by calculating the ratio of two sides of a right-angled triangle.
- \(\sin{x} = \frac{\text{opposite}}{\text{hypotenuse}}\)
- \(\cos{x} = \frac{\text{adjacent}}{\text{hypotenuse}}\)
- \(\tan{x} = \frac{\text{opposite}}{\text{adjacent}}\)
A useful way to remember these is:
\(s^o_h~c^a_h~t^o_a\)
Exact trigonometric ratios for 0°, 30°, 45°, 60° and 90°
The trigonometric ratios for the angles 30°, 45° and 60° can be found using two special triangles.
An equilateral triangle with side lengths of 2 cm can be used to find exact values for the trigonometric ratios of 30° and 60°.
The equilateral triangle can be split into two right-angled triangles.
Using either of these right-angled triangles, Pythagoras can be used to find the third side of the right-angled triangle.
A right-angled isosceles triangle with two sides of length 1 cm can be used to find exact values for the trigonometric ratios of 45°.
Calculate the length of the third side of the triangle using Pythagoras' theorem.
The exact trigonometric ratios for 0°, 30°, 45°, 60° and 90° are:
| \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | \(90^\circ\) | |
| \(\sin{x}\) | \(0\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(\cos{x}\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) |
| \(\tan{x}\) | \(0\) | \(\frac{1}{\sqrt{3}}~\text{or}~\frac{\sqrt{3}}{3}\) | \(1\) | \(\sqrt{3}\) | \(\text{Undefined}\) |
| \(\sin{x}\) | |
|---|---|
| \(0^\circ\) | \(0\) |
| \(30^\circ\) | \(\frac{1}{2}\) |
| \(45^\circ\) | \(\frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) |
| \(60^\circ\) | \(\frac{\sqrt{3}}{2}\) |
| \(90^\circ\) | \(1\) |
| \(\cos{x}\) | |
|---|---|
| \(0^\circ\) | \(1\) |
| \(30^\circ\) | \(\frac{\sqrt{3}}{2}\) |
| \(45^\circ\) | \(\frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\) |
| \(60^\circ\) | \(\frac{1}{2}\) |
| \(90^\circ\) | \(0\) |
| \(\tan{x}\) | |
|---|---|
| \(0^\circ\) | \(0\) |
| \(30^\circ\) | \(\frac{1}{\sqrt{3}}~\text{or}~\frac{\sqrt{3}}{3}\) |
| \(45^\circ\) | \(1\) |
| \(60^\circ\) | \(\sqrt{3}\) |
| \(90^\circ\) | \(\text{Undefined}\) |
\(\tan{90}\) is undefined because \(\tan{90} = \frac{1}{0}\) and division by zero is undefined (a calculator will give an error message).